>> Lecturer: Let's look at the graphs of some logarithmic functions and we'll do this from the perspective that they are inverses of exponential functions. Now earlier, we generated the inverse of this exponential function. We know all about its graph. Its graph is right here. The inverse we found to be Y equals log base 2 of X. And we can designate that as F inverse of X, then is log base 2 of X. Now, we know over here that we have a graph then, of this logarithmic function that is a reflection in this Y equals X line. So it'll be a graph that will make its appearance over here. Now, if we were going to generate a table of values for the construction of the graph, we could do that, but using our calculator would be a little difficult because we have a logarithm in base 2. And right now, we don't have a way to use the calculator to calculate the logarithm of numbers in base 2. We'll learn about that in a little bit, but we don't know about it right now. But there is another technique that we can use, and it's the technique of using the notion of that inverse idea. That is, I have a table of values here associated with the exponential function. And that's -- these points were plotted to get this graph. All right, all I have to do to generate the graph of the inverse is to simply swap the role of X and Y here. So I can make a table of values for the inverse function just by taking the X column here and writing it as the Y column and the Y column here and writing it as the X column here and then plot the points. One-fourth goes to negative 2. Well, let's see. An X of one-fourth goes to negative 2. One-half goes to negative 1; 1, 0; 2, 1; 4, 2, and 8, 3. And the graph. And this is about what we would expect. Now, notice also that we started with this exponential function whose domain is -- well let's see. I'm going to write the domain and range of that function. We want to investigate those ideas as well for the logarithmic function. So the domain for our exponential function, the acceptable replacements for X algebraically or graphically -- it's for what values of X do we have points on the graph? Well for all values of X we have points on the graph, so the domain would be all real numbers. I'm going to use this designation for the set of real numbers. How about range? Well, the range -- the Y values that are on the graph. For our exponential function now, the Y values would be the Y values that lie up here. None of these Y values have X partners that put points on the graph. So it's just Y values that are distinctly positive, because a Y of 0 doesn't appear on the graph either because we have an X of -- excuse me -- an asymptote at the X axis. So it would be Y values that are greater than 0. Now notice what happens with the logarithmic function. The roles of the domain and range are simply swapped. So for domain -- for the logarithmic function, the domain is -- let's see. For what Xs do we have points on the graph? Well, for all of these Xs that are positive, we have points on the graph. So it would be X greater than 0. What about range? For the range, it would be -- let's see. For what values of Y do we have points on the graph? Well, for all values of Y, we have points on the graph, even for the Ys up here. The X coordinates for Ys that lie way up here would be way out that way. But at any rate, they do exist. So the range would be all real numbers. Now also the asymptotes have kind of swapped positions in a sense as well. They have actually reflected in our Y equals X line too. And we can't see them because they are the axes that are on our graph. They are the axes for the coordinate plane. But the X axis is the asymptote for our exponential function. And then when we do this swap around, this reflection, it turns out that the Y axis is the asymptote for our logarithmic function. Well, we can examine the graph of the whole family of logarithmic functions by using this approach that these functions are inverses of exponential functions. For example, we might want to look at the shape of the graph of log X. Now, that's log understood base 10.